with some \(c\ne 0\)  and hence \((0,0,{\bar{\tau }},\eta )\)  is effectively hyperbolic characteristic of \(p_2\) . Then following [1]}, [2]} there is a parametrix of \({\hat{P}}_2\)  with finite propagation speed of micro supports. Since \({\hat{P}}_1\)  is a first order operator with real \(p_1\)  it is easy to see that \({\hat{P}}_1\)  has a parametrix with finite propagation speed of micro supports. Therefore \(P\)  has a parametrix with finite propagation speed of micro supports. Turn to the case that \((0,0,{\bar{\tau }},\eta )\)  is not a critical point. Writing \(p_2\)  as
\(p_2(t, x,\tau ,\xi )=\tau ^2-a(t, x,\xi )|\xi |^2\) 
